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Difference between revisions of "Complete uniform space"

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A [[Uniform space|uniform space]] in which every [[Cauchy filter|Cauchy filter]] converges. An important example is a [[Complete metric space|complete metric space]]. A closed subspace of a complete uniform space is complete; a complete subspace of a separable uniform space is closed. The product of complete uniform spaces is complete; conversely, if the product of non-empty uniform spaces is complete, then all the spaces are complete. Any uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023930/c0239301.png" /> can be uniformly and continuously mapped onto some dense subspace of a complete uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023930/c0239302.png" /> (see [[Completion of a uniform space|Completion of a uniform space]]).
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====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR></table>
 
 
 
 
 
 
 
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A [[Uniform space|uniform space]] in which every [[Cauchy filter|Cauchy filter]] converges. An important example is a [[Complete metric space|complete metric space]]. A closed subspace of a complete uniform space is complete; a complete subspace of a separable uniform space is closed. The product of complete uniform spaces is complete; conversely, if the product of non-empty uniform spaces is complete, then all the spaces are complete. Any uniform space $X$ can be uniformly and continuously mapped onto some dense subspace of a complete uniform space $\hat{X}$ (see [[Completion of a uniform space|Completion of a uniform space]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"J.R. Isbell,  "Uniform spaces" , Amer. Math. Soc.  (1964)</TD></TR></table>
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|valign="top"|{{Ref|Bo}}|| N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)
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|valign="top"|{{Ref|Is}}|| J.R. Isbell,  "Uniform spaces" , Amer. Math. Soc.  (1964)
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|valign="top"|{{Ref|Ke}}|| J.L. Kelley,    "General topology" , Springer  (1975)
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Latest revision as of 17:31, 9 December 2013

2020 Mathematics Subject Classification: Primary: 54E15 Secondary: 54E50 [MSN][ZBL]

A uniform space in which every Cauchy filter converges. An important example is a complete metric space. A closed subspace of a complete uniform space is complete; a complete subspace of a separable uniform space is closed. The product of complete uniform spaces is complete; conversely, if the product of non-empty uniform spaces is complete, then all the spaces are complete. Any uniform space $X$ can be uniformly and continuously mapped onto some dense subspace of a complete uniform space $\hat{X}$ (see Completion of a uniform space).

References

[Bo] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
[Is] J.R. Isbell, "Uniform spaces" , Amer. Math. Soc. (1964)
[Ke] J.L. Kelley, "General topology" , Springer (1975)
How to Cite This Entry:
Complete uniform space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_uniform_space&oldid=30895
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article